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 Uspekhi Mat. Nauk, 2010, Volume 65, Issue 1(391), Pages 145–176 (Mi umn9345)

Closed 1-forms in topology and geometric group theory

M. Farbera, R. Geogheganb, D. Schütza

a University of Durham, UK
b State University of New York, Binghamton, USA

Abstract: In this article we describe relations of the topology of closed 1-forms to the group-theoretic invariants of Bieri–Neumann–Strebel–Renz. Starting with a survey, we extend these Sigma invariants to finite CW-complexes and show that many properties of the group-theoretic version have analogous statements. In particular, we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik–Schnirelmann category of a closed 1-form and to the existence of a non-singular closed 1-form in a given cohomology class on a high-dimensional closed manifold.
Bibliography: 32 titles.

Keywords: Sigma invariants, Lusternik–Schnirelmann category, Novikov ring, movability of homology classes.

DOI: https://doi.org/10.4213/rm9345

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English version:
Russian Mathematical Surveys, 2010, 65:1, 143–172

Bibliographic databases:

UDC: 515.165
MSC: Primary 20J05, 58E05; Secondary 55M30

Citation: M. Farber, R. Geoghegan, D. Schütz, “Closed 1-forms in topology and geometric group theory”, Uspekhi Mat. Nauk, 65:1(391) (2010), 145–176; Russian Math. Surveys, 65:1 (2010), 143–172

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn9345
• https://doi.org/10.4213/rm9345
• http://mi.mathnet.ru/eng/umn/v65/i1/p145

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Suciu A.I., “Fundamental groups, Alexander invariants, and cohomology jumping loci”, Topology of algebraic varieties and singularities, Contemp. Math., 538, Amer. Math. Soc., Providence, RI, 2011, 179–223
2. T. Hüttemann, D. Quinn, “Finite domination and Novikov rings: Laurent polynomial rings in two variables”, J. Algebra Appl., 14:4 (2015), 1550055, 44 pp.
3. Dowdall S., Kapovich I., Leininger Ch.J., “Dynamics on Free-By-Cyclic Groups”, 19, no. 5, 2015, 2801–2899
4. Friedl S., Maxim L., “Twisted Novikov homology of complex hypersurface complements”, Math. Nachr., 290:4 (2017), 604–612
5. Dowdall S., Kapovich I., Leininger Ch.J., “Mcmullen Polynomials and Lipschitz Flows For Free-By-Cyclic Groups”, J. Eur. Math. Soc., 19:11 (2017), 3253–3353
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